notebooks/22. Data Mining II/22.d. Data Mining II- Silhouette Coefficients.ipynb

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  10    "cells": [
  11     {
  12      "cell_type": "heading",
  13      "level": 1,
  14      "metadata": {},
  15      "source": [
  16       "Section 22.d. Data mining II: Silhouette Coefficients"
  17      ]
  18     },
  19     {
  20      "cell_type": "markdown",
  21      "metadata": {},
  22      "source": [
  23       "In this notebook, we will investigate how silhouette coefficients are used to select an appropriate value $k$ for $k$-means clustering."
  24      ]
  25     },
  26     {
  27      "cell_type": "code",
  28      "collapsed": false,
  29      "input": [
  30       "import numpy as np\n",
  31       "import matplotlib.pyplot as plt\n",
  32       "\n",
  33       "import pandas as pd\n",
  34       "\n",
  35       "%pylab inline\n",
  36       "\n",
  37       "np.set_printoptions(precision=2, linewidth=100)"
  38      ],
  39      "language": "python",
  40      "metadata": {},
  41      "outputs": [
  42       {
  43        "output_type": "stream",
  44        "stream": "stdout",
  45        "text": [
  46         "Populating the interactive namespace from numpy and matplotlib\n"
  47        ]
  48       },
  49       {
  50        "output_type": "stream",
  51        "stream": "stderr",
  52        "text": [
  53         "/Users/agw96/virtualenv_environments/tm351/lib/python3.3/site-packages/pandas/io/excel.py:626: UserWarning: Installed openpyxl is not supported at this time. Use >=1.6.1 and <2.0.0.\n",
  54         "  .format(openpyxl_compat.start_ver, openpyxl_compat.stop_ver))\n"
  55        ]
  56       }
  57      ],
  58      "prompt_number": 1
  59     },
  60     {
  61      "cell_type": "heading",
  62      "level": 2,
  63      "metadata": {},
  64      "source": [
  65       "Silhouette Coefficient"
  66      ]
  67     },
  68     {
  69      "cell_type": "markdown",
  70      "metadata": {},
  71      "source": [
  72       "The following function definition implements the silhouette coefficient as defined in section 22.6 of the module notes. That is:\n",
  73       "    \n",
  74       "$\\hspace{5em}s(i)=\\frac{b(i)-a(i)}{max(a(i), b(i))}$\n",
  75       "\n",
  76       "where $a(i)$ is the average distance of data instance $i$ to data instances within a cluster, and $b(i)$ is the average distance of data instance $i$ to data instances in other clusters. \n",
  77       "\n",
  78       "The function is called with two variables: `silhouette(X,cIDX)`. The input data instances, `X` are a numpy array, and `cIDX` is the resulting cluster labels of the data instances.\n"
  79      ]
  80     },
  81     {
  82      "cell_type": "code",
  83      "collapsed": false,
  84      "input": [
  85       "from scipy.spatial.distance import pdist, squareform\n",
  86       "from matplotlib import cm\n",
  87       "    \n",
  88       "def silhouette(X, cIDX):\n",
  89       "    \"\"\"\n",
  90       "    Computes the silhouette score for each instance of a clustered dataset,\n",
  91       "    which is defined as:\n",
  92       "        s(i) = (b(i)-a(i)) / max{a(i),b(i)}\n",
  93       "    with:\n",
  94       "        -1 <= s(i) <= 1\n",
  95       "\n",
  96       "    Args:\n",
  97       "        X    : A M-by-N array of M observations in N dimensions\n",
  98       "        cIDX : array of len M containing cluster indices (starting from zero)\n",
  99       "\n",
 100       "    Returns:\n",
 101       "        s    : silhouette value of each observation\n",
 102       "    \"\"\"\n",
 103       "\n",
 104       "    N = X.shape[0]              # number of instances\n",
 105       "    K = len(np.unique(cIDX))    # number of clusters\n",
 106       "\n",
 107       "    # compute pairwise distance matrix\n",
 108       "    D = squareform(pdist(X))\n",
 109       "\n",
 110       "    # indices belonging to each cluster\n",
 111       "    kIndices = [np.flatnonzero(cIDX==k) for k in range(K)]\n",
 112       "\n",
 113       "    # compute a,b,s for each instance\n",
 114       "    a = np.zeros(N)\n",
 115       "    b = np.zeros(N)\n",
 116       "    for i in range(N):\n",
 117       "        # instances in same cluster other than instance itself\n",
 118       "        a[i] = np.mean( [D[i][ind] for ind in kIndices[cIDX[i]] if ind!=i] )\n",
 119       "        # instances in other clusters, one cluster at a time\n",
 120       "        b[i] = np.min( [np.mean(D[i][ind]) for k,ind in enumerate(kIndices) if cIDX[i]!=k] )\n",
 121       "    s = (b-a)/np.maximum(a,b)\n",
 122       "\n",
 123       "    # plot\n",
 124       "    order = np.lexsort((-s,cIDX))\n",
 125       "    indices = [np.flatnonzero(cIDX[order]==k) for k in range(K)]\n",
 126       "    ytick = [(np.max(ind)+np.min(ind))/2 for ind in indices]\n",
 127       "    ytickLabels = [\"%d\" % x for x in range(K)]\n",
 128       "    cmap = cm.jet( np.linspace(0,1,K) ).tolist()\n",
 129       "    clr = [cmap[i] for i in cIDX[order]]\n",
 130       "\n",
 131       "    fig = plt.figure()\n",
 132       "    ax = fig.add_subplot(111)\n",
 133       "    ax.barh(range(X.shape[0]), s[order], height=1.0, edgecolor='none', color=clr)\n",
 134       "    ax.set_ylim(ax.get_ylim()[::-1])\n",
 135       "    plt.yticks(ytick, ytickLabels)\n",
 136       "    plt.xlabel('Silhouette Coefficient')\n",
 137       "    plt.ylabel('Cluster')\n",
 138       "    \n",
 139       "    return s"
 140      ],
 141      "language": "python",
 142      "metadata": {},
 143      "outputs": [],
 144      "prompt_number": 2
 145     },
 146     {
 147      "cell_type": "markdown",
 148      "metadata": {},
 149      "source": [
 150       "For example, we calculate Silhouette coefficient of the k-means clustering result with $k = 2$ for the example data given in Figure 22.3 using the `silhouette` function. The function generates a plot that shows Silhouette coefficient for each of the data instances in an order for each cluster and prints the Silhouette coefficient for each data instance. A higher Silhouette coefficient means a better clustered data instance. \n"
 151      ]
 152     },
 153     {
 154      "cell_type": "markdown",
 155      "metadata": {},
 156      "source": [
 157       "First, reimplement the data from Figure 22.3 from the course notes:"
 158      ]
 159     },
 160     {
 161      "cell_type": "code",
 162      "collapsed": false,
 163      "input": [
 164       "\n",
 165       "# Data\n",
 166       "data = pd.DataFrame({'Attribute 1':[4, 2, 1, 1, 4, 5, 8, 9, 6, 8],\n",
 167       "                     'Attribute 2':[8, 7, 5, 6, 5, 2, 3, 2, 1, 4]},\n",
 168       "                     index=['A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J'])\n",
 169       "data"
 170      ],
 171      "language": "python",
 172      "metadata": {},
 173      "outputs": [
 174       {
 175        "html": [
 176         "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
 177         "<table border=\"1\" class=\"dataframe\">\n",
 178         "  <thead>\n",
 179         "    <tr style=\"text-align: right;\">\n",
 180         "      <th></th>\n",
 181         "      <th>Attribute 1</th>\n",
 182         "      <th>Attribute 2</th>\n",
 183         "    </tr>\n",
 184         "  </thead>\n",
 185         "  <tbody>\n",
 186         "    <tr>\n",
 187         "      <th>A</th>\n",
 188         "      <td> 4</td>\n",
 189         "      <td> 8</td>\n",
 190         "    </tr>\n",
 191         "    <tr>\n",
 192         "      <th>B</th>\n",
 193         "      <td> 2</td>\n",
 194         "      <td> 7</td>\n",
 195         "    </tr>\n",
 196         "    <tr>\n",
 197         "      <th>C</th>\n",
 198         "      <td> 1</td>\n",
 199         "      <td> 5</td>\n",
 200         "    </tr>\n",
 201         "    <tr>\n",
 202         "      <th>D</th>\n",
 203         "      <td> 1</td>\n",
 204         "      <td> 6</td>\n",
 205         "    </tr>\n",
 206         "    <tr>\n",
 207         "      <th>E</th>\n",
 208         "      <td> 4</td>\n",
 209         "      <td> 5</td>\n",
 210         "    </tr>\n",
 211         "    <tr>\n",
 212         "      <th>F</th>\n",
 213         "      <td> 5</td>\n",
 214         "      <td> 2</td>\n",
 215         "    </tr>\n",
 216         "    <tr>\n",
 217         "      <th>G</th>\n",
 218         "      <td> 8</td>\n",
 219         "      <td> 3</td>\n",
 220         "    </tr>\n",
 221         "    <tr>\n",
 222         "      <th>H</th>\n",
 223         "      <td> 9</td>\n",
 224         "      <td> 2</td>\n",
 225         "    </tr>\n",
 226         "    <tr>\n",
 227         "      <th>I</th>\n",
 228         "      <td> 6</td>\n",
 229         "      <td> 1</td>\n",
 230         "    </tr>\n",
 231         "    <tr>\n",
 232         "      <th>J</th>\n",
 233         "      <td> 8</td>\n",
 234         "      <td> 4</td>\n",
 235         "    </tr>\n",
 236         "  </tbody>\n",
 237         "</table>\n",
 238         "</div>"
 239        ],
 240        "metadata": {},
 241        "output_type": "pyout",
 242        "prompt_number": 3,
 243        "text": [
 244         "   Attribute 1  Attribute 2\n",
 245         "A            4            8\n",
 246         "B            2            7\n",
 247         "C            1            5\n",
 248         "D            1            6\n",
 249         "E            4            5\n",
 250         "F            5            2\n",
 251         "G            8            3\n",
 252         "H            9            2\n",
 253         "I            6            1\n",
 254         "J            8            4"
 255        ]
 256       }
 257      ],
 258      "prompt_number": 3
 259     },
 260     {
 261      "cell_type": "markdown",
 262      "metadata": {},
 263      "source": [
 264       "Next, run the k-means algorithm on the data with $k=2$. This will show how well the clustering works for two clusters:"
 265      ]
 266     },
 267     {
 268      "cell_type": "code",
 269      "collapsed": false,
 270      "input": [
 271       "import sklearn.cluster as sc\n",
 272       "\n",
 273       "# Run k-means algorithm on the data (k=2)\n",
 274       "kmeans = sc.KMeans(n_clusters=2)\n",
 275       "assigned_clusters = kmeans.fit(data).labels_\n",
 276       "\n",
 277       "silhouette(data, assigned_clusters)"
 278      ],
 279      "language": "python",
 280      "metadata": {},
 281      "outputs": [
 282       {
 283        "metadata": {},
 284        "output_type": "pyout",
 285        "prompt_number": 4,
 286        "text": [
 287         "array([ 0.51,  0.69,  0.62,  0.69,  0.32,  0.41,  0.68,  0.66,  0.58,  0.58])"
 288        ]
 289       },
 290       {
 291        "metadata": {},
 292        "output_type": "display_data",
 293        "png": 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LFixocV/aesn/8ccfW69eveydd95JUsraecl+7Ngxq66uNjOzvXv3Ws+ePZMRtVb1+eyY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\njqDwAcAR/x9bFW6trbXoGQAAAABJRU5ErkJggg==\n",
 294        "text": [
 295         "<matplotlib.figure.Figure at 0x105d2cb90>"
 296        ]
 297       }
 298      ],
 299      "prompt_number": 4
 300     },
 301     {
 302      "cell_type": "heading",
 303      "level": 3,
 304      "metadata": {},
 305      "source": [
 306       "Exercise 22.5"
 307      ]
 308     },
 309     {
 310      "cell_type": "markdown",
 311      "metadata": {},
 312      "source": [
 313       "Measure the cluster validity using the Silhouette coefficient for the k-means algorithm using $k = 2$ and $k=3$ for the English Premier League data. \n",
 314       "\n",
 315       "The data is reproduced here:"
 316      ]
 317     },
 318     {
 319      "cell_type": "code",
 320      "collapsed": false,
 321      "input": [
 322       "premierLeague=pd.DataFrame([(68, 41), (39, 61), (32, 74), (71, 27),\n",
 323       "                            (33, 48), (61, 39), (40, 85), (38, 53), \n",
 324       "                            (101, 50), (102, 37), (64, 43), (43, 59), \n",
 325       "                            (28, 62), (54, 46), (45, 52), (41, 60), \n",
 326       "                            (54, 54), (55, 51), (43, 59), (40, 51)],\n",
 327       "                           index=['Arsenal', 'Aston Villa', 'Cardiff City', 'Chelsea', \n",
 328       "                                  'Crystal Palace', 'Everton', 'Fulham', 'Hull City', \n",
 329       "                                  'Liverpool', 'Manchester City', 'Manchester United',\n",
 330       "                                  'Newcastle United', 'Norwich City', 'Southampton',\n",
 331       "                                  'Stoke City', 'Sunderland', 'Swansea City',\n",
 332       "                                  'Tottenham Hotspur', 'West Bromwich Albion',\n",
 333       "                                  'West Ham United'],\n",
 334       "                           columns=['Goals for', 'Goals against'])\n",
 335       "\n",
 336       "premierLeague"
 337      ],
 338      "language": "python",
 339      "metadata": {},
 340      "outputs": [
 341       {
 342        "html": [
 343         "<div style=\"max-height:1000px;max-width:1500px;overflow:auto;\">\n",
 344         "<table border=\"1\" class=\"dataframe\">\n",
 345         "  <thead>\n",
 346         "    <tr style=\"text-align: right;\">\n",
 347         "      <th></th>\n",
 348         "      <th>Goals for</th>\n",
 349         "      <th>Goals against</th>\n",
 350         "    </tr>\n",
 351         "  </thead>\n",
 352         "  <tbody>\n",
 353         "    <tr>\n",
 354         "      <th>Arsenal</th>\n",
 355         "      <td>  68</td>\n",
 356         "      <td> 41</td>\n",
 357         "    </tr>\n",
 358         "    <tr>\n",
 359         "      <th>Aston Villa</th>\n",
 360         "      <td>  39</td>\n",
 361         "      <td> 61</td>\n",
 362         "    </tr>\n",
 363         "    <tr>\n",
 364         "      <th>Cardiff City</th>\n",
 365         "      <td>  32</td>\n",
 366         "      <td> 74</td>\n",
 367         "    </tr>\n",
 368         "    <tr>\n",
 369         "      <th>Chelsea</th>\n",
 370         "      <td>  71</td>\n",
 371         "      <td> 27</td>\n",
 372         "    </tr>\n",
 373         "    <tr>\n",
 374         "      <th>Crystal Palace</th>\n",
 375         "      <td>  33</td>\n",
 376         "      <td> 48</td>\n",
 377         "    </tr>\n",
 378         "    <tr>\n",
 379         "      <th>Everton</th>\n",
 380         "      <td>  61</td>\n",
 381         "      <td> 39</td>\n",
 382         "    </tr>\n",
 383         "    <tr>\n",
 384         "      <th>Fulham</th>\n",
 385         "      <td>  40</td>\n",
 386         "      <td> 85</td>\n",
 387         "    </tr>\n",
 388         "    <tr>\n",
 389         "      <th>Hull City</th>\n",
 390         "      <td>  38</td>\n",
 391         "      <td> 53</td>\n",
 392         "    </tr>\n",
 393         "    <tr>\n",
 394         "      <th>Liverpool</th>\n",
 395         "      <td> 101</td>\n",
 396         "      <td> 50</td>\n",
 397         "    </tr>\n",
 398         "    <tr>\n",
 399         "      <th>Manchester City</th>\n",
 400         "      <td> 102</td>\n",
 401         "      <td> 37</td>\n",
 402         "    </tr>\n",
 403         "    <tr>\n",
 404         "      <th>Manchester United</th>\n",
 405         "      <td>  64</td>\n",
 406         "      <td> 43</td>\n",
 407         "    </tr>\n",
 408         "    <tr>\n",
 409         "      <th>Newcastle United</th>\n",
 410         "      <td>  43</td>\n",
 411         "      <td> 59</td>\n",
 412         "    </tr>\n",
 413         "    <tr>\n",
 414         "      <th>Norwich City</th>\n",
 415         "      <td>  28</td>\n",
 416         "      <td> 62</td>\n",
 417         "    </tr>\n",
 418         "    <tr>\n",
 419         "      <th>Southampton</th>\n",
 420         "      <td>  54</td>\n",
 421         "      <td> 46</td>\n",
 422         "    </tr>\n",
 423         "    <tr>\n",
 424         "      <th>Stoke City</th>\n",
 425         "      <td>  45</td>\n",
 426         "      <td> 52</td>\n",
 427         "    </tr>\n",
 428         "    <tr>\n",
 429         "      <th>Sunderland</th>\n",
 430         "      <td>  41</td>\n",
 431         "      <td> 60</td>\n",
 432         "    </tr>\n",
 433         "    <tr>\n",
 434         "      <th>Swansea City</th>\n",
 435         "      <td>  54</td>\n",
 436         "      <td> 54</td>\n",
 437         "    </tr>\n",
 438         "    <tr>\n",
 439         "      <th>Tottenham Hotspur</th>\n",
 440         "      <td>  55</td>\n",
 441         "      <td> 51</td>\n",
 442         "    </tr>\n",
 443         "    <tr>\n",
 444         "      <th>West Bromwich Albion</th>\n",
 445         "      <td>  43</td>\n",
 446         "      <td> 59</td>\n",
 447         "    </tr>\n",
 448         "    <tr>\n",
 449         "      <th>West Ham United</th>\n",
 450         "      <td>  40</td>\n",
 451         "      <td> 51</td>\n",
 452         "    </tr>\n",
 453         "  </tbody>\n",
 454         "</table>\n",
 455         "</div>"
 456        ],
 457        "metadata": {},
 458        "output_type": "pyout",
 459        "prompt_number": 5,
 460        "text": [
 461         "                      Goals for  Goals against\n",
 462         "Arsenal                      68             41\n",
 463         "Aston Villa                  39             61\n",
 464         "Cardiff City                 32             74\n",
 465         "Chelsea                      71             27\n",
 466         "Crystal Palace               33             48\n",
 467         "Everton                      61             39\n",
 468         "Fulham                       40             85\n",
 469         "Hull City                    38             53\n",
 470         "Liverpool                   101             50\n",
 471         "Manchester City             102             37\n",
 472         "Manchester United            64             43\n",
 473         "Newcastle United             43             59\n",
 474         "Norwich City                 28             62\n",
 475         "Southampton                  54             46\n",
 476         "Stoke City                   45             52\n",
 477         "Sunderland                   41             60\n",
 478         "Swansea City                 54             54\n",
 479         "Tottenham Hotspur            55             51\n",
 480         "West Bromwich Albion         43             59\n",
 481         "West Ham United              40             51"
 482        ]
 483       }
 484      ],
 485      "prompt_number": 5
 486     },
 487     {
 488      "cell_type": "heading",
 489      "level": 4,
 490      "metadata": {},
 491      "source": [
 492       "First, calculate the coefficints for $k = 2$."
 493      ]
 494     },
 495     {
 496      "cell_type": "code",
 497      "collapsed": false,
 498      "input": [
 499       "\n",
 500       "# Run k-means algorithm on the data (k=2)\n",
 501       "kmeans = sc.KMeans(n_clusters=2)\n",
 502       "assigned_clusters = kmeans.fit(premierLeague).labels_\n",
 503       "\n",
 504       "silhouette(premierLeague, assigned_clusters)"
 505      ],
 506      "language": "python",
 507      "metadata": {},
 508      "outputs": [
 509       {
 510        "metadata": {},
 511        "output_type": "pyout",
 512        "prompt_number": 6,
 513        "text": [
 514         "array([ 0.41,  0.73,  0.62,  0.46,  0.62,  0.21,  0.51,  0.7 ,  0.46,  0.5 ,  0.26,  0.72,  0.66,\n",
 515         "        0.24,  0.64,  0.73,  0.44,  0.34,  0.72,  0.68])"
 516        ]
 517       },
 518       {
 519        "metadata": {},
 520        "output_type": "display_data",
 521        "png": 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3ryZNmqTVq1erVatWiYwoqfbns2/fvrp69apKSkoSEU9SfBl37typMWPGqEOHDlqxYoUm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\nA4Aj/j9524DP6slcNQAAAABJRU5ErkJggg==\n",
 522        "text": [
 523         "<matplotlib.figure.Figure at 0x106fc5910>"
 524        ]
 525       }
 526      ],
 527      "prompt_number": 6
 528     },
 529     {
 530      "cell_type": "heading",
 531      "level": 4,
 532      "metadata": {},
 533      "source": [
 534       "Similarly, calculate the coefficients for $k = 3$:"
 535      ]
 536     },
 537     {
 538      "cell_type": "code",
 539      "collapsed": false,
 540      "input": [
 541       "from sklearn.cluster import KMeans\n",
 542       "\n",
 543       "# Run k-means algorithm on the data (k=3)\n",
 544       "kmeans = KMeans(n_clusters=3)\n",
 545       "assigned_clusters = kmeans.fit(premierLeague).labels_\n",
 546       "\n",
 547       "silhouette(premierLeague, assigned_clusters)"
 548      ],
 549      "language": "python",
 550      "metadata": {},
 551      "outputs": [
 552       {
 553        "metadata": {},
 554        "output_type": "pyout",
 555        "prompt_number": 7,
 556        "text": [
 557         "array([ 0.63,  0.64,  0.55,  0.37,  0.44,  0.65,  0.41,  0.55,  0.69,  0.69,  0.66,  0.57,  0.59,\n",
 558         "        0.45,  0.28,  0.62,  0.17,  0.36,  0.57,  0.46])"
 559        ]
 560       },
 561       {
 562        "metadata": {},
 563        "output_type": "display_data",
 564        "png": 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 565        "text": [
 566         "<matplotlib.figure.Figure at 0x10b665890>"
 567        ]
 568       }
 569      ],
 570      "prompt_number": 7
 571     },
 572     {
 573      "cell_type": "code",
 574      "collapsed": false,
 575      "input": [],
 576      "language": "python",
 577      "metadata": {},
 578      "outputs": [],
 579      "prompt_number": 7
 580     }
 581    ],
 582    "metadata": {}
 583   }
 584  ]
 585 }